91Ó°ÊÓ

The distributive property is a fundamental principle in algebra that allows you to simplify expressions. It states that for any numbers or expressions, \(a(b + c) = ab + ac\). This property lets us multiply a term outside the parentheses by each term inside the parentheses, breaking down complex problems into manageable parts.

In the context of our problem, the distributive property helps us multiply \(2a\) by each term inside the parentheses: \(a^2 - 2b + 6\). This means you distribute \(2a\) to each of these terms one by one.

• First, multiply \(2a\) by \(a^2\), yielding \(2a^3\).
• Next, multiply \(2a\) by \(-2b\), resulting in \(-4ab\).
• Finally, multiply \(2a\) by \(6\), which gives \(12a\).

By applying the distributive property, complex expressions are broken into smaller algebraic products, each of which can be simplified individually. Mastering this technique is essential for tackling more advanced algebra problems.

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They don't always have an equal sign, which distinguishes them from equations. In our exercise, the expression inside the parentheses, \(a^2 - 2b + 6\), is an example of an algebraic expression.

Understanding algebraic expressions is crucial as they form the foundational blocks of algebra. In our problem, we work with:

• Variables: Letters such as \(a\) and \(b\) that represent numbers.
• Coefficients: Numbers like \(2\) that are multiplied by the variables.
• Constants: Stand-alone numbers, such as \(6\), in the expression.

Algebraic expressions can be simplified using rules like the distributive property and by combining like terms. Mastery of manipulating these expressions is necessary for solving equations and real-world problems.

A monomial is a type of algebraic expression that consists of only one term. This term can be a number, a variable, or a product of numbers and variables raised to a positive integer power. Monomials are the simplest forms of algebraic expressions and appear in our distribution steps when \(2a\) is multiplied with each term.

In the given exercise, after applying the distributive property, each resulting part, such as \(2a^3\), \(-4ab\), and \(12a\), is a monomial.

• 2a³: Made by multiplying \(2a\) by \(a^2\), showing how variables and exponents combine in monomials.
• -4ab: Formed by multiplying \(2a\) with \(-2b\), combining coefficient multiplication and variable multiplication.
• 12a: Simply the product of \(2a\) and \(6\), a basic form of a monomial.

Monomials can be added, subtracted, multiplied, and factored to simplify and solve algebraic expressions, making them integral to understanding algebraic operations.